Front Matter

Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information Lectures on Algebraic Cycles Second Edition Spencer Bloch’s 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch–Kato conjecture on special values of zeta functions. The book begins with Mumford’s example showing that the Chow group of zero-cycles on an algebraic variety can be infinite dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena. spencer bloch is R. M. Hutchins Distinguished Service Professor in the Department of Mathematics at the University of Chicago. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information NEW MATHEMATICAL MONOGRAPHS Editorial Board Bela´ Bollobas´ William Fulton Anatole Katok Frances Kirwan Peter Sarnak Barry Simon Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/series/sSeries.asp?code=NMM 1 M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups 2 J. B. Garnett and D. E. Marshall Harmonic Measure 3 P. Cohn Free Ideal Rings and Localization in General Rings 4 E. Bombieri and W. Gubler Heights in Diophantine Geometry 5 Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs 6 S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures 7 A. Shlapentokh Hilbert’s Tenth Problem 8 G. Michler Theory of Finite Simple Groups I 9 A. Baker and G. Wustholz¨ Logarithmic Forms and Diophantine Geometry 10 P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds 11 B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) 12 J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory 13 M. Grandis Directed Algebraic Topology 14 G. Michler Theory of Finite Simple Groups II 15 R. Schertz Complex Multiplication © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information Lectures on Algebraic Cycles Second Edition SPENCER BLOCH University of Chicago © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao˜ Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521118422 C S. Bloch 2010 C Mathematics Department, Duke University 1980 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First edition published in 1980 by Duke University, Durham, NC 27706, USA Second edition published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-11842-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information Contents Preface to the second edition page vii 0 Introduction 3 1 Zero-cycles on surfaces 9 Appendix: On an argument of Mumford in the theory of algebraic cycles 21 2 Curves on threefolds and intermediate jacobians 25 3 Curves on threefolds – the relative case 37 4 K-theoretic and cohomological methods 45 5 Torsion in the Chow group 59 2 6 Complements on H (K2) 69 7 Diophantine questions 79 8 Relative cycles and zeta functions 97 9 Relative cycles and zeta functions – continued 115 Bibliography 123 Index 129 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information Preface to the second edition 30 Years later... Looking back over these lectures, given at Duke University in 1979, I can say with some pride that they contain early hints of a number of important themes in modern arithmetic geometry. Of course, the flip side of that coin is that they are now, thirty years later, seriously out of date. To bring them up to date would involve writing several more monographs, a task best left to mathematicians thirty years younger than me. What I propose instead is to comment fairly briefly on several of the lectures in an attempt to put the reader in touch with what I believe are the most important modern ideas in these areas. The section on motives just below is intended as a brief introduction to the modern viewpoint on that subject. The remaining sections until the last follow roughly the content of the original book, though the titles have changed slightly to reflect my current emphasis. The last section, motives in physics, represents my recent research. In the original volume I included a quote from Charlie Chan, the great Chi- nese detective, who told his bumbling number one son “answer simple, but question very very hard.” It seemed to me an appropriate comment on the subject of algebraic cycles. Given the amazing deep new ideas introduced into the subject in recent years, however, I think now that the question remains very very hard, but the answer is perhaps no longer so simple... At the end of this essay I include a brief bibliography, which is by no means complete. It is only intended to illustrate the various ideas mentioned in the text. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information viii Preface to the second edition Motives Much of the recent work in this area is centered around motives and the construction – in fact various constructions, due to Hanamura (1995), Levine (1998), and Voevodsky (Mazza et al. 2006; Voevodsky et al. 2000) – of a trian- gulated category of mixed motives. I will sketch Voevodsky’s construction as it also plays a central role in his proof of the Bloch–Kato–Milnor conjecture discussed in Lecture 5. Then I will discuss various lectures from the original monograph. Let k be a field. The category Cork is an additive category with objects = → smooth k-varieties: Ob(Cork) Ob(Smk). Morphisms Z : X Y are finite linear combinations of correspondences Z = niZi where Zi ⊂ X × Y is closed and the projection πi : Zi → X is finite and surjective. Intuitively, we may think → n ∈ −1 of Zi as a map X Sym Y associating to x X the fibre fi (x)viewedasa zero-cycle on Y. There is an evident functor Smk → Cork which is the identity on objects. A presheaf on a category C with values in an abelian category A is simply a op contravariant functor F : C →A.AnA-valued presheaf F on Cork induces | a presheaf F Smk on Smk. Intuitively, to lift a presheaf G from Smk to Cork one needs a structure of trace maps or transfers f∗ : G(Z) → G(X)forZ/X finite. Presheaves on Cork are referred to as presheaves with transfers. For X ∈ Ob(Smk) one has the representable sheaf Ztr(X) defined by = , . Ztr(X)(U) HomCork (U X) An important elaboration on this idea yields for pointed objects xi ∈ Xi Ztr((X1, x1) ∧···∧(Xn, xi)) := Coker Ztr(X1 ×···Xi ×···Xn) → Ztr( Xi) . n 1 In particular, one defines Ztr( Gm) by taking Xi = A −{0} and xi = 1. A presheaf with transfers F is called homotopy invariant if, with obvious ∗ = ∗ × 1 → notation, i0 i1 : F(U A ) F(U). The complex of chains C∗(F)ona presheaf with transfers F is the presheaf of complexes (placed in cohomological degrees [−∞, 0]) n 0 C∗(F):= U →···→ F(U × ∆ ) →···→ F(U × ∆ ) Here n (0.1) ∆ := Spec k[t0,...,tn]/( ti − 1) © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information Preface to the second edition ix is the algebro-geometric n-simplex. The boundary maps in the complex are the usual alternating sums of restrictions to the faces ∆n−1 → ∆n defined by setting ti = 0. The two restrictions ∗, ∗ × 1 → i0 i1 : C∗(F)(U A ) C∗(F)(U) are shown to be homotopic, so the homology presheaves Hn(C∗(F)) are homotopy invariant. 1 1 Maps f0, f1 : X → Y in Cork are A -homotopic if there exists H : X×A → Y = ∗ 1 1 in Cork such that f j i j H.

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